This choice of \(\beta \)  is closely linked to robust scale estimators (see [1]}) and in particular Huber's second proposal [2]} which also makes use of \(\psi ^2\)  to find \(\beta \) . In this article, we limit ourselves to the simple estimation of location estimator and not of simultaneous estimation of scale and location as it is done for Huber's second proposal, doing so could be an extension of this work.
